Properties

Degree $2$
Conductor $25200$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 4·11-s − 6·13-s + 4·17-s + 2·19-s + 8·23-s − 6·29-s − 4·31-s + 8·37-s + 2·41-s + 6·43-s − 6·47-s + 49-s + 2·53-s − 4·59-s + 14·67-s + 2·71-s − 6·73-s + 4·77-s − 8·79-s − 8·83-s − 6·89-s + 6·91-s − 18·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.20·11-s − 1.66·13-s + 0.970·17-s + 0.458·19-s + 1.66·23-s − 1.11·29-s − 0.718·31-s + 1.31·37-s + 0.312·41-s + 0.914·43-s − 0.875·47-s + 1/7·49-s + 0.274·53-s − 0.520·59-s + 1.71·67-s + 0.237·71-s − 0.702·73-s + 0.455·77-s − 0.900·79-s − 0.878·83-s − 0.635·89-s + 0.628·91-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25200\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{25200} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 25200,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.52812657057285, −15.08194719304134, −14.57837060466167, −14.17591533525934, −13.31036035500617, −12.80004050058433, −12.65472209858211, −11.91472192166041, −11.14465171090973, −10.88610109319816, −9.905788912277660, −9.783539165757425, −9.201962524867983, −8.396306023612352, −7.626313880742181, −7.409301669367557, −6.862187215943042, −5.818370850315464, −5.408229622891023, −4.905172202938420, −4.161161575842946, −3.152097799900458, −2.819898187149391, −2.053587037063464, −0.9272685895166880, 0, 0.9272685895166880, 2.053587037063464, 2.819898187149391, 3.152097799900458, 4.161161575842946, 4.905172202938420, 5.408229622891023, 5.818370850315464, 6.862187215943042, 7.409301669367557, 7.626313880742181, 8.396306023612352, 9.201962524867983, 9.783539165757425, 9.905788912277660, 10.88610109319816, 11.14465171090973, 11.91472192166041, 12.65472209858211, 12.80004050058433, 13.31036035500617, 14.17591533525934, 14.57837060466167, 15.08194719304134, 15.52812657057285

Graph of the $Z$-function along the critical line